Sufficient condition such that weak and initial topology coincide for a locally convex space

Question

This is the opposite question to this one: Example of locally convex space such that its weak and initial topology coincide.

If we have a normed vector space $X$ than its norm topology and weak topology coincide if and only if $X$ is finite dimensional. Now I'm interested in the general case of $X$ being just locally convex.

If $X$ is a locally convex space (which is not normed) what are sufficient conditions such that the weak topology and the initial topology of $X$ do not coincide?

I think that there must be at least one quite general condition, since I think that it is rather rare that we have a locally convex space fulfilling the condition that its weak and initial topology coincide.

Answer 1

The following is mentioned in the book by H.Schaefer (IV, Exercise 6(a)), but with a little bit different formulation:

Theorem. A complete locally convex space $X$ over $\mathbb K$ ($\mathbb K=\mathbb R$ or $\mathbb C$) has weak topology (i.e. the topology of $X$ coincides with the weak topology on $X$ generated by all linear continuous functionals $f:X\to{\mathbb K}$) if and only if $X$ is isomorphic (as a locally convex space) to the direct product of a family of copies of $\mathbb K$: $$ X\cong{\mathbb K}^{\mathfrak m} $$ for some cardinality ${\mathfrak m}$.

Corollary. If $X$ is complete and $X\not\cong{\mathbb K}^{\mathfrak m}$, then the topology of $X$ can't be weak. In particular, a Fréchet space $X$ has weak topology if and only if $X\cong{\mathbb K}^{\mathbb N}$.

Answer 2

EDIT: Have just realised that I misunderstood your question. The first paragraph is stil valid and I am leaving the last three paragraphs to show where I misunderstood you. I thought that under initial topology you meant the finest l.c. topology on $E$.

An infinite dimensional metrisable l.c.s. is never complete for the weak topology. So any complete, infinite dimensional space wii do what you want (assuming I have now understood your question correctly). Sorry for the confusion.

THE SOLUTION TO MY UNDERSTANDING OF YOUR QUESTION. Each vector space has a finest l.c. topology. It can be defined as that l.c. structure which is defined by ALL seminorms on $E$ or as the l.c. inductive limit of the the finite dimensional subspaces. An example of an l.c. space for which this coincides with its weak topology is $\phi$, the space of finite sequences, with the natural inductive limit structure. Its topological and algebraic dual is $\omega$, the space of ALL sequences. For this see G. Köthe: Topological linear spaces.

If a l.c. space has a non continuous linear functional, then the above two topologies cannot coincide for the simple reason that they have distinct duals.

A host of such spaces can be found as follows: a separable l.c.s. whose linear dimension is uncountable automatically has a non continuous linear functional.

The last fact uses AC in the form of the existence of a Hamel basis. Work of Solovay, Schwartz and Garnir shows that without AC the situation is different.